Exposing the Gambler's Fallacy
You're playing roulette, and red has just come up eight times in a row! Now's the time to bet on black, right? Wrong. Red and black are equally likely. Even though we've just seen a bunch of reds. How can this be? Let's find out.
The idea that an event becomes more likely if it hasn't happened for a while is called the gambler's fallacy. (It applies to non-gambling events too, though.) It seems logical, but it's not true. If it were true, all you'd have to do to win would be to wait until red has come up several times in a row and bet on black. If that worked, casinos couldn't stay in business. But because it doesn't work, they're happy to provide the marquee that shows recent numbers, and the pads and pencils or baccarat players to track what's happened before. That's your clue that it's not gonna help.
The reason it doesn't work is simple: it doesn't matter what happened before. Past results have no bearing on future results. If the odds of getting heads on a coin flip are 50-50, it doesn't matter what was flipped before. Same for the roulette wheel. But coin flips are easy to understand, so let's look at those.
The probability of getting heads on a coin flip is 1 out of 2. That's one way to win out of two possible outcomes. We can write that as the fraction 1/2. That's easy, but what about the odds of getting two heads in a row, on two flips? To figure this we multiply by the probability of each event:
Okay, so what are the chances of getting ten heads in a row?
x x x x x x x x x =
That's 1 in 1,024. Not very likely, of course.
So here's where the gambler's fallacy comes in: Say you've tossed the coin nine times and amazingly, you got nine heads. You figure that the next toss will be tails, because the probability of getting ten heads in a row is one in 1024, which is really unlikely to happen.
The problem with this reasoning is that you're not looking at the chances of getting ten heads in a row, you're looking at the chances of getting one heads in a row. The heads that already happened no longer have a 50% chance of happening, they already happened. When you flip again the odds for that flip will be 50-50, same as it ever was.
Let's introduce our hero, Mr. P ("P" for "probability"). He'll always be looking to the future to see what's going to happen. He's about to make ten coin flips, hoping to get ten heads. Here's his outlook:
Mr. P, lucky guy that he is, got nine heads in a row. Here he is, getting ready to make his tenth flip:
Now you're saying, Hey, wait! How come all the 1/2's turned into 1's? The answer is that they're no longer unknowns. Before you flip a coin you don't know what's going to happen so you have 50-50 odds. But after you flip the coin you definitely know what happened! After you flip a coin, the probability that you got a result is 1. You definitely flipped the coin. Definitely, definitely. So after you've flipped nine heads, the probability of flipping a tenth head is 1x1x1x1x1x1x1x1x1x 1/2 = 1/2.
Let's have another look at Mr. P:
Notice that it doesn't matter where on the line you stick him, the chances of his next flip being heads is always 1/2. Wherever he is, it doesn't matter what happened before, his chances on his next toss are always 1 in 2.
How could it be otherwise? When you flip a coin you will get one result out of two possible outcomes. That's 1 in 2, or 1/2. Why and how could those numbers change just because you got a bunch of heads or tails already? They couldn't. The coin has no memory, it neither knows nor cares what was flipped before. If it's a 1-out-of-2 coin, it will always be a 1-out-of-2 coin.
Still not convinced? Then here's another way to think about it: Let's say someone hands you a coin and asks, "What are the chances of flipping heads?" Without hesitation you'd probably say 1 out of 2? But wait a minute—if it were true that heads were more likely if tails has just come up a bunch of times, then why did you answer "1 in 2" right away when asked about the chances of getting heads? Why didn't you say, "Well, first you have to tell me whether tails has been coming up a lot before I can tell you whether heads has a fair shot or not."? It's simple: You didn't ask about the previous flips because intuitively you know they're unimportant. If someone hands you a coin, the chances of getting heads are 1 in 2, regardless of what happened before.
Would it really be the case that you answered "1 in 2," and then your friend said, "Oh, I forgot to tell you, tails has just come up nine times in a row." Would you now suddenly change your answer and say that heads is more likely? I hope not.
One last example: Let's say your friend slides two quarters towards you across the table. She tells you that the first coin has been flipping normally, but the second quarter has just had nine tails in a row. Would you now believe that the chances of getting heads on the first coin are even but the chances of getting heads on the second coin are greater? Given two identical coins, could you really believe that one would be more likely to flip heads than the other? I hope not!
The same concept applies to roulette. An American roulette wheel has 18 red spots, 18 black spots, and 2 green spots. The chances of getting red on any one spin are 18/38. If you just saw nine reds in a row, what is the likelihood of getting black on the next spin?
18/38, same as it ever was.